Exploring the Borderland between Physics and Chemistry: Theoretical Methods in the Study of Atomic Clusters - Metallic Systems: A Quantum Chemist's Perspective - page 377

identiied. at. the. lower. level.. There. are. a. variety. of. stochastic. methodologies,. and.
each.has.several.variants..We.will.briely.describe.three.of.them:.genetic.algorithm,.
simulated.annealing,.and.the.big.bang.method.
11.3.1 g enetic a lgorithm
Genetic.algorithm.techniques.are.based.on.the.ideas.of.Darwinian.biological.evo-
lution. and. have. recently. been. widely. used. in. the. optimization. of. atomic. clusters.
[23,24]..One.starts.by.deining.a. genome .or.string.that.represents.a.candidate.solu-
tion.to.the.problem..In.the.case.of.atomic.clusters,.the.choice.is.simple:.the.genome.is.
a.set.of.coordinates.for.each.atom.of.the.cluster..One.starts.with.an.initial.population.
of.genomes.selected.randomly..This.population.will.be.propagated.to.produce.more.
“it”.species.by.applying.“natural.selection”.rules;.in.the.case.of.clusters.this.means.
combining.clusters.with.low.energies.to.(hopefully).produce.new.cluster.geometries,.
which.lead.to.still.lower.cluster.energies..In.order.to.increase.sampling.of.the.search.
space,. one. deines. operators. simulating. crossover. and. mutation. and. applies. these.
to.the.developing.population..The.lowest.energy.clusters.are.produced.based.on.the.
principle.of.“survival.of.the.ittest.”.At.each.step,.population.members.with.energy.
above.a.given.threshold.are.deleted.from.the.population,.and.species.with.low.ener-
gies.are.allowed.to.reproduce..In.this.way,.after.a.given.number.of.generations,.one.
should.obtain.structures.of.lower.energy..In.the.case.of.clusters,.this.means.low-lying.
isomers.
The.initial.population.of.individuals,.represented.as.a.set.of.atomic.coordinates,.
is.generated.randomly..In.practice,.different.constraints.can.be.imposed.to.avoid.
searching. very. unphysical. regions. of. the. hypersurface.. The. coordinates. should.
be.constrained.to.lie.inside.a.box.of.the.expected.dimensions.of.the.cluster,.and.
any. pair. of. atoms. can. neither. be. closer. than. a. given. distance. nor. separated. by.
more.than.a.given.distance..The.speciic.criteria.for.desired.itness,.selection.rules,.
and. crossover. and. mutation. probabilities. are. speciic. to. every. genetic. algorithm.
implementation.. In. this. part. of. the. work,. we. followed. the. algorithm. presented.
in. Ref.. [25].. The. number. of. times. one. needs. to. evaluate. the. energy. can. become.
signiicantly.large..Therefore,.it.is.a.common.practice.to.produce.all.generations.
in.the.genetic.algorithm.cycles.using.an.empirical.molecular.orbital.method,.such.
as.MSINDO.[26]..It.is.also.important.to.note.that.each.individual.is.an.optimized.
geometry. within. the. semiempirical. method.. When. the. genetic. algorithm. opti-
mization. is. complete,. the. resulting. isomers. are. re-optimized. using. a. high-level.
electronic.structure.theory,.usually.Kohn-Sham.DFT.including.an.exchange-cor-
relation.functional.via.a.conventional.gradient-following.quasi-Newton.optimiza-
tion.technique.
As.an.example,.in.Ref..[22].the.genetic.algorithm.was.used.to.ind.14.isomers.of.the.
Si 9 .cluster..The.eight.most.important.isomer.structures.are.presented.in.Figure.11.2..
All.of.the.structures.were.optimized.using.the.B3PW91.functional.with.the.Stuttgart.
pseudopotential.[27]..Since.the.corresponding.basis.set.does.not.contain.diffuse.and.
polarization.functions,.the.basis.was.augmented.with.diffuse.s-.and.p-functions.and.
one.set.of.d-polarization.function.from.the.Sadlej.basis.set.[28]..To.ensure.that.the.
optimized.structures.are.stationary.points.on.the.molecular.potential.energy.surface,.